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The invasion of Indo-Pacific lionfish (Pterois volitans) together with unsustainable harvesting of herbivorous fish has caused major ecological changes in the western Atlantic and Caribbean coral reefs. Developing an effective harvesting policy of herbivorous reef fish and the invasive lionfish is necessary to minimize the adverse ecological impacts. To evaluate the potential efficacy of controlling the growth of lionfish population, we have proposed a non-smooth two-stage-structured mathematical model with two harvesting policies (non-selective and selective) of lionfish determined by a control function with a discontinuity at a critical herbivorous fish density. We study the sliding mode dynamics of the Filippov system, the boundary equilibrium bifurcation, and the pseudo-saddle-node bifurcation. We use threshold harvesting with hysteresis that helps in preventing the collapse of the herbivorous fish population through a bounded oscillation about pseudo-equilibrium. Our results show that a longer herbivorous fish harvesting period can be achieved in selective harvesting policy without compromising the herbivorous fish stock.

Bistability, the presence of alternative stable states, is an important feature of population models as it indicates that long-term predictions are dependent on the current population density. Two distinct kinds of bistability re-occur in population modelling studies, Allee Bistability and Positive Bistability. In this article, we show that a novel codimension-3 bifurcation, the cusp-transcritical interaction, can act as an organising centre for ordinary differential equations that exhibit both Allee Bistability and Positive Bistability. We first show how a normal form for cusp-transcritical interactions emerges from the unfolding of a particular one-dimensional degeneracy. We then illustrate the ecological relevance of the cusp-transcritical interaction. Finally, we provide a comprehensive example of normal-form analysis of an existing population model that demonstrates the occurrence of the codimension-3 bifurcation. We note that Allee Bistability and Positive Bistability may manifest unexpectedly in complex, ecological models, and therefore, this bifurcation-focused approach can provide valuable insight into the behaviour of newly developed ecosystem models.

Human and robotic legged locomotion can be described with complex multi-degree-of-freedom dynamic models, whose bifurcation or parameter analysis may explain some features of typical patterns during motion. In this paper, we focus on the effect of kinematic parameters and foot placement techniques on the ground-foot impact intensity. The work is based on a multibody dynamic model of a segmented leg, which possesses some fundamental characteristics of locomotion systems: (a) distinct topology in the flight and ground phases; (b) kinetic energy absorption due to partially/fully inelastic ground-foot collision; (c) an active control strategy for maintaining a prescribed mechanical energy level; and (d) different control strategies for the flight and ground phases. We obtain a quantitative measure for the foot collision intensity by analytic calculations. The pre-impact velocity conditions are obtained from a hopping three-segmented planar leg model that imitates pedal locomotion. The single-legged model contains the foot, the shank, the thigh and a reaction wheel attached to the hip, which models the effect of the upper body. The existence of stable periodic motion associated with hopping is shown for a wide range of parameters by means of finding suitable control torques in the ankle, the knee and the hip joint. The parameters of the linear feedback controller are tuned to optimize different cost functions, such as running speed, energy efficiency and impact intensity. We also investigate how the stability of the periodic motion depends on the control gains.

In this chapter, we show how the conflict between the shareholders (owners) and managers of firms in terms of profit rates generates dynamics between growth and distribution that results in a long-run variation in the capacity utilization rate. The model developed here generates oscillations in the rate of capacity utilization in the short run before settling down to its long-run value. Furthermore, the long-run value of the rate of capacity utilization falls within a range of plausible values, and this range is determined by the conflict between shareholders and managers. The conflict as a closure, we believe, provides a more realistic microeconomic underpinning to study the impact of distribution on accumulation and long-run utilization. In doing so, we have not taken the approach of the existence of normal utilization rate that is relied upon by the Harrodian authors and the endogenization of animal spirits in such a way that the actual utilization influences the desired or normal rate of utilization by the Kaleckian authors. The model yields hysteresis in that it generates two different disequilibrium growth paths when shareholders and managers struggle to gain control of the firm.

The effects of small-amplitude additive Gaussian white noise on the one-dimensional square root map are investigated. In particular, the focus is on the unexpected effects noise of varying amplitudes has on the system for parameter regions just outside intervals of multistability. It is shown that in these regions periodic behaviour that is unstable in the deterministic system can be effectively stabilised by the addition of noise of an appropriate amplitude. Features of noise-induced transitions from stable to stabilised unstable periodic behaviour are highlighted, and it is shown how these features can be understood by examining relative levels of expansion and contraction in the deterministic map.

In this paper we describe the complex structure of the basins of attraction of stable periodic orbits of the one-dimensional square root map and how this produces sensitivity to the addition of small amplitude noise. In particular we focus on how noise of varying amplitudes affects the system in parameter regions of attractor coexistence and also how trajectories jump between different periodic behaviours. We show that there is a non-monotonic relationship between the noise amplitude and the proportion of time spent in each periodic behaviour. These relationships are explained by examining approximations of steady-state distributions of trajectory deviations due to noise and the complicated deterministic structures of the map. We also show how the effect of noise scales on consecutive intervals of multistability.

The concept of density-dependent population growth is fundamental to our understanding of how populations persist. While it is generally agreed that negative density dependence must occur at high densities, the direction of density dependence may be negative (pure negative density dependence) or positive (demographic Allee effect) at low densities. In this article, we present a technique to link the direction of density dependence to generic ecological factors. This technique involves exploiting the presence of a particular bifurcation, known as a saddle-node-transcritical interaction. We first provide a method to detect this bifurcation in a given model and then demonstrate its ecological relevance using several existing mechanistic models. With a mathematical framework in place, we are able to identify scenarios in which neither a weak Allee effect nor pure negative density dependence are possible. More generally, we find conditions on parameter values that are necessary for transitions between pure negative density dependence and demographic Allee effects to occur.

This paper focuses on a hybrid adaptive control (or communication protocol) that is used to drive a network of agents towards consensus. We develop an approximation to the system to capture the qualitative behaviour of the most distant node state and gain dynamics, thus serving as a powerful reduction of order of the larger system. Numerical results are presented to argue the robustness of the approximation over a range of differing initial network sizes and distributions of initial state values.

In many animal species, recruitment is facilitated by adults’ efforts to protect offspring from predation. Theoretical studies of this phenomenon have usually focused on resolving the conflict between an individual's self-preservation and its attempts to successfully reproduce. While the decision to protect is made at the level of a single individual, the aggregation of these decisions may affect population density and structure. This idea motivates the development of a functional response for predators that is compatible with the protective behaviour of prey. We use this functional response to study the long-term behaviour of a protective prey population under different levels of predation. We find that contribution of protective effort may promote or inhibit population density depending on the riskiness associated with interference. Moreover, our results suggest that, in environments characterised by intense predation, a protection-driven Allee effect allows sufficiently large populations to persist. We interpret these results in the context of different strategies for newborn defence.

The discrete element method typically uses an explicit numerical integration scheme to solve the equations of motion. However, like all explicit schemes, the scheme is only conditionally stable, with the stability determined by the size of the time-step. Currently, there are no comprehensive techniques for estimating appropriate DEM time-steps when a nonlinear contact interaction is used. It is common practice to apply a large factor of safety to these estimates to ensure stability which unnecessarily increases the computational cost of these simulations. This work introduces an alternative framework for selecting a stable time-step for nonlinear contact laws, specifically for the Hertz–Mindlin contact law. This approach uses the fact that the discretised equations of motion take the form of a nonlinear map and can be analysed as such. Using this framework, we analyse the effects of both system damping and the initial relative velocity of collision on the critical time-step for a Hertz–Mindlin contact event between spherical particles.

Interspecific interactions may occur for just a brief period each year before the populations involved become spatially separated. For instance, the range of a migrating population may overlap with that of a population of predators for a single season. In this work, we outline a framework for examining how this kind of ‘transient’ predation influences the dynamics of the prey population. A time-dependent switching system is used to partition the annual cycle into distinct segments. We then consider the effect of a single predatory interaction during a particular season, with the associated predators characterised as either generalists or specialists. We show that generalist predation potentially can allow multiple stable limit cycles to exist. Predation by specialists may cause prey abundance to oscillate over long time periods. This is shown to be a consequence of over-exploitation of newborn prey individuals. The habitat-based formulation extends naturally to the study of interannual variation in environmental conditions. We illustrate how such changes may cause migrant populations to undergo sudden changes in numbers that are not readily reversible.

For stability and bifurcation analysis involving recurrent behaviour such as periodic orbits, it is important to be able to quantify how nearby trajectories behave by means of a local mapping. In smooth systems these mappings can be computed using the system’s variational equations. For piecewise-smooth or hybrid systems the same technique cannot be used without some corrections. This is due to the fact that nearby trajectories can be topologically distinct because they can undergo different sequences of events associated with the system’s discontinuity boundaries. To account for this, one can derive zero-time discontinuity mappings associated with boundary interactions. In this paper we derive zero-time discontinuity mappings for piecewise-smooth vector fields and hybrid dynamical systems in which the position of the discontinuity boundary has a stochastic component. In particular, we consider systems with stochastically oscillating boundaries and systems with stochastic imperfections on the discontinuity boundary.

In a smooth dynamical system the characteristics of a given reference trajectory can be determined, to lowest order, by examining the linearised system about the reference trajectory. In other words, we can approximate the deviations of trajectories after a given time, with starting points in a neighbourhood of the reference trajectory, by multiplying the initial deviations by the corresponding fundamental matrix solution. This form of analysis cannot be used directly in nonsmooth systems as the vector field is either not everywhere differentiable or the flow function is not continuous. To account for this, one can derive the zero-time discontinuity mapping (ZDM) associated with the discontinuity boundary. The Jacobian of this mapping is known as the saltation matrix and its properties can tell us how the crossing of the discontinuity boundary affects the deviations of trajectories from a reference trajectory. In particular, this matrix can be composed with the fundamental matrix solutions of the individual flows on either side of the discontinuity boundary in order to determine the overall fundamental matrix solution of a trajectory that crosses the boundary. In this paper we derive a saltation matrix for a piecewise-smooth dynamical system in which the position of the discontinuity boundary oscillates according to a mean-reverting stochastic process. The derived saltation matrix contains the entire effect of both the discontinuity and the uncertainty introduced into the system by the noisy boundary, and is composable with the deterministic fundamental matrix solutions of the individual flows to give the overall fundamental matrix solution of a crossing trajectory. We also present some simple examples of piecewise-smooth systems with stochastically varying boundaries, analysed using the derived noisy saltation matrix. In particular we focus on the analysis of a discontinuous variant of the Chua circuit. In this case we apply noise to the system’s discontinuity boundaries which are generated by the piecewise-linear nature of the voltage–current response of the Chua diode. We find that our method allows us to analyse the effects of boundary noise on periodic attractors close to bifurcation points. In particular we show that we can use the method to accurately predict the noise amplitudes required to destroy or merge periodic attractors.

The breeding success of many species depends on the alignment of the period of maximum demand of offspring with a period of food abundance. In this paper, we use piecewise-smooth differential equations to model the variation in the size of a population that has a short interval each year during which successful reproduction is possible. An increase in spring-time temperatures in recent decades has caused these intervals to advance, leading to temporal mismatches between species on different trophic levels. We first consider a one-species model which illustrates the dynamics of a population of specialist feeders over the course of a single breeding season and use it to examine how reproductive success depends on the population's distribution of breeding dates. We then introduce time-dependent switches to extend the model to a broader class of species. Steady-state solutions are used to measure the extent to which a migratory population can sustain itself over long time scales and repeated breeding events. The model's predictions agree with the observed negative effect on population size associated with an advance in the food season if not adequately compensated for by the population. Finally, we discuss how this model can be extended in order to encompass a wider range of ecological systems and to highlight the mechanisms over which climate change may exert an influence.

In this article, we will introduce the phenomenon known as the Painlevé paradox and further discuss the associated coupled phenomena, jam and lift-off. We analyze under what conditions the Painlevé paradox can occur for a general two-body collision using a framework that can be easily used with a variety of impact laws, however, in order to visualize jam and lift-off in a numerical simulation, we choose to use a recently developed energetic impact law as it is capable of achieving a unique forward solution in time. Further, we will use this framework to derive the criteria under which the Painlevé paradox can occur in a forced double-pendulum mechanical system. First, using a graphical technique, we will show that it is possible to achieve the Painlevé paradox for relatively low coefficient of friction values, and second we will use the energetic impact law to numerically show the occurrence of the Painlevé paradox in the double-pendulum system.

Centromere Protein A (CENP-A) is a histone H3 variant found at mammalian centromeres. Unlike canonical histones which are incorporated at centromeres in S phase, CENP-A is deposited at centromeric chromatin in G1. Although recent studies have elucidated many of the molecular details associated with the CENP-A incorporation pathway, some aspects of the process are still not fully understood. CENP-A incorporation in G1 requires multiple assembly factors for its recruitment and maintenance. In this study, the first mathematical model of the CENP-A incorporation pathway is developed. The model is based on what is currently known about the pathway and is calibrated by comparing numerical simulations with experimental observations taken from the literature. The model succinctly collates a large body of knowledge accumulated in recent decades concerning the pathway and produces results that are consistent with experimental findings. It identifies possible gaps in what is currently known about the pathway and suggests possible directions for future research. It is envisaged that the model will be expanded upon and improved as more information concerning the pathway comes to light.